In the following set of pedagogical posts ("lectures"), I will try to address a few intriguing aspects about the Universe, notably about it's topology and possible implications thereof.

If you are interested in more details about the topology of the Universe and possibilities to observe effects of topology, don't miss the following link to a well readable paper from Cambridge U.(UK)/Princeton U./U. Cleveland:

The geometry is a more or less LOCAL attribute of the space, while the topology tells us about its global shape. General relativity relates the mass distribution of the universe to its geometry and of course, the geometry of the universe determines the dynamics of the mass! General relativity does not at all constrain the topology of the universe.

Topology also tells us whether the space is singly or multiply connected.

What does the latter mean?

Flat space (e.g. a plane) is simply connected, meaning there is only one direct path for light to travel from a source to an observer. But the universe might instead be "multiply connected," like a doughnut (torus), in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd!!!

Note how we can make a doughnut from a plane (sheet of paper):

We simply glue together the opposite sides of the sheet: first glue together two to form a cylinder and then you glue the two ends of the cylinder together.

If you place yourself anywhere on the surface of the doughnut you can easily see that light emitted from a source can reach you along different paths along the doughnut's surface! Since the speed of light is finite, you'll see multiple images of the source corresponding to (vastly) different times!

(see the above article for more details and illustrations)

Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact, be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps more than once--creating multiple images of each galaxy.

This latter point is highly intriguing, though: since the universe is really big, light would take billions of years to go around once and thus the different images of the same galaxy would picture it in entirely different stages of its evolution!! So when looking for mirror images of galaxies this has to be crucially taken into account.

Bye Fridger

PS:
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Topology is a fascinating and most important discipline in mathematics. It deals with characterizing apparently different geometrical shapes in terms of simple common attributes. So you learn that a cube and a sphere are topologically the same thing. They may be deformed into each other. A doughnut is fundamentally different from a sphere, though! It's equivalent to a sphere with a hole! That additional hole is CRUCIAL: on the surface of a sphere, EVERY inscribed loop may be contracted to a point, while this is impossible on a doughnut (why?? , example? specify a loop that cannot be contracted to a point! )

Along these lines you may easily find out the basic differences between a teapot and a doghnut, for example

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Last edited by t00fri on Sun, 09-09-07, 21:29 GMT, edited 2 times in total.

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Notably if one factor f1(r) -> 0 tends to zero while the other one tends to infinity f2(r) -> infinity, while the independent variable r becomes large, r -> infinity++++++++++++++++++++

Let us consider a simple example:

Let d(r) be the number density of UFOs , i.e. their number within a 3d spacial sphere of radius r around the observer. Suppose we knew how that UFO-density depends on the distance r from the observer,

e.g.

Code:

d(r) = log(1+ 10/r^3)

Clearly, if r increases towards infinity, d(r) tends to ZERO, since the argument of the logarithm tends to 1 and log(1) = 0.

So if we ask about the density of UFOs in an INFINITE Universe (r -> infinity) the answer is ZERO.

But suppose we want to extract the TOTAL NUMBER of UFOs in the Universe from the function d(r), we must multiply the number density d(r) with the

Code:

(spherical) spacial volume V(r) = 4/3 Pi r^3.

Code:

N(r) = d(r) * V(r)

then tells us the number of UFOs in a spacial volume of radius r. In the limit r-> infinity, N(infinity) then gives the desired result for the whole INFINITE UNIVERSE.

Is N(infinity) ZERO, INFINITE or FINITE?

The tricky point is that

Code:

d(r) -> 0 while V(r) -> infinity for large r!

How does the PRODUCT d*V behave?

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What is 0 * infinity in this case?
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A priory the product 0*infinity is undefined. It could have any value. But for our example we can uniquely find the answer as follows:

We must write down explicity how d(r) behaves very close to where it vanishes, i.e. for VERY large r. For that purpose we can expand the logarithm in d(r) in form of a series of terms each one vanishing more rapidly than the previous one, like so
There are explicit algorithms how to find such a series expansion for a given function. Maple knows of course how to do this .

If you plot and compare the exact expression and the approximate right hand side you will find that it represents a VERY good approximation of d(r) for large enough values of r.

Now we may just multiply the series term by term with the Volume = 4/3* Pi * r^3 and perform the limit r -> infinity. Then, obviously, all terms but the first one vanish.

The result for the total number of UFOs in the infinite Universe is therefore

Now that you have become topology experts I think it's about time to illuminate a bit the breathtaking possibilities arising from a possible finite, yet multiply connected spacial topology of our Universe

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We proceed VERY slowly and intuitively!
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Let me start from a 2d Universe first since there we can still draw easily what we mean.

The simplest finite and geometrically flat, singly connected 2d Universe is the blue square you can see below on the top left (Fig 1). In this flat universe the Pythagoras (triangle) theorem would clearly hold.

An observer in the red galaxy can perceive the light emitted from the yellow galaxy only along ONE path, the yellow one in Fig. 1)! This corresponds to the fact that our assumed space topology so far is singly connected.

Now, let us make a multiply connected 2d space out of our finite-sized Universe "sheet"! Abstractly speaking this happens if we just identify the opposite sides of our flat rectangle. Look e.g. at the white path of light emitted from the yellow galaxy! It can now reach our observer in the red galaxy: when the light hits the right-hand vertical border of our universe in Fig. 1) it reappears immediately at the opposite left-hand border as depicted (on account of the border identification). Eventually the white path hits the observer's eyes! Analogously, the red path and the top-bottom borders...

This identification of the opposite borders of our finite square Universe may sound somewhat artificial to you at first, but it is NOT at all.

Look at Figs. 2) and 3) what it actually implies. By identifying the opposite borders we have just made a nice doghnut shaped surface (space) out of our original flat rectangle! Figs. 2) and 3) illustrate the steps intuitively. You immediately see how the white yellow and red paths of light from the yellow galaxy can be perceived by our observer in the red one. That 2d doghnut Universe is now multiply connected and correspondingly more than 1 image of the yellow galaxy can be seen by the observer. Since the distance s also differ, the different galaxy images correspond to largely different evolution stages of our yellow galaxy, given the long light travelling times involved.

Of course, the doghnut surface represents a curved 2d space unlike the rectangle we have started with!
Correspondingly the Pythagoras theorem does not hold anymore for triangles inscribed on the doghnut surface.

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Next we increase the spacial dimension of our finite universe by 1 and realistically consider 3d. Now it's really becoming fun!
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Instead of our 2d sheet in Fig. 1) we now start from a finite cubic 3d Universe as shown here

Analogously to the procedure above , let us now make a 3d-doughnut (torus) out of the cube by identifying the opposite walls of the cube. This is indicated by the matching colors in the figure.

Unfortunately a 3d-doghnut cannot be drawn anymore but conceptionally everything is as in the 2d example above.

Still we may intuitively proceed: The identification may be simulated by replacing our identified walls by mirrors. You, the observer now stands in the center of a finite-sized mirror hall, which quite a few of you must have experienced in reality already.

To you the finite multiply connected 3d doghnut Universe appears as if it was infinite, since through the mirrors you perceive in all directions infinitely many copies of yourself and of the finite cubic "base" Universe. By looking towards the right-hand mirror wall you may e.g. see the back of your own head!

That seems to be a good point to let you contemplate a bit about the amazing implications. Please don't hesitate to ask if there is something unclear at this point.

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